TPTP Problem File: ALG232+2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ALG232+2 : TPTP v8.2.0. Released v3.4.0.
% Domain : General Algebra
% Problem : Algebraic Operation on Subsets of Many Sorted Sets T12
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Mar97] Marasik (1997), Algebraic Operation on Subsets of Many
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t13_closure3 [Urb08]
% Status : Theorem
% Rating : 0.97 v8.1.0, 0.94 v7.5.0, 1.00 v7.3.0, 0.97 v7.1.0, 0.96 v7.0.0, 0.97 v6.4.0, 0.92 v6.3.0, 0.96 v6.2.0, 0.92 v6.1.0, 0.93 v6.0.0, 0.96 v5.5.0, 1.00 v4.0.1, 0.96 v3.7.0, 1.00 v3.4.0
% Syntax : Number of formulae : 4740 (1086 unt; 0 def)
% Number of atoms : 21718 (3162 equ)
% Maximal formula atoms : 49 ( 4 avg)
% Number of connectives : 19126 (2148 ~; 171 |;8405 &)
% ( 711 <=>;7691 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 6 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 342 ( 340 usr; 1 prp; 0-4 aty)
% Number of functors : 774 ( 774 usr; 263 con; 0-9 aty)
% Number of variables : 11032 (10488 !; 544 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Bushy version: includes all articles that contribute axioms to the
% Normal version.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
% : Infinox says this has no finite (counter-) models.
%------------------------------------------------------------------------------
include('Axioms/SET007/SET007+0.ax').
include('Axioms/SET007/SET007+1.ax').
include('Axioms/SET007/SET007+2.ax').
include('Axioms/SET007/SET007+3.ax').
include('Axioms/SET007/SET007+4.ax').
include('Axioms/SET007/SET007+6.ax').
include('Axioms/SET007/SET007+7.ax').
include('Axioms/SET007/SET007+8.ax').
include('Axioms/SET007/SET007+9.ax').
include('Axioms/SET007/SET007+10.ax').
include('Axioms/SET007/SET007+11.ax').
include('Axioms/SET007/SET007+14.ax').
include('Axioms/SET007/SET007+16.ax').
include('Axioms/SET007/SET007+17.ax').
include('Axioms/SET007/SET007+20.ax').
include('Axioms/SET007/SET007+24.ax').
include('Axioms/SET007/SET007+25.ax').
include('Axioms/SET007/SET007+26.ax').
include('Axioms/SET007/SET007+31.ax').
include('Axioms/SET007/SET007+34.ax').
include('Axioms/SET007/SET007+35.ax').
include('Axioms/SET007/SET007+40.ax').
include('Axioms/SET007/SET007+48.ax').
include('Axioms/SET007/SET007+51.ax').
include('Axioms/SET007/SET007+54.ax').
include('Axioms/SET007/SET007+55.ax').
include('Axioms/SET007/SET007+64.ax').
include('Axioms/SET007/SET007+67.ax').
include('Axioms/SET007/SET007+80.ax').
include('Axioms/SET007/SET007+86.ax').
include('Axioms/SET007/SET007+117.ax').
include('Axioms/SET007/SET007+125.ax').
include('Axioms/SET007/SET007+182.ax').
include('Axioms/SET007/SET007+186.ax').
include('Axioms/SET007/SET007+188.ax').
include('Axioms/SET007/SET007+200.ax').
include('Axioms/SET007/SET007+205.ax').
include('Axioms/SET007/SET007+210.ax').
include('Axioms/SET007/SET007+212.ax').
include('Axioms/SET007/SET007+213.ax').
include('Axioms/SET007/SET007+225.ax').
include('Axioms/SET007/SET007+363.ax').
include('Axioms/SET007/SET007+393.ax').
include('Axioms/SET007/SET007+395.ax').
include('Axioms/SET007/SET007+396.ax').
include('Axioms/SET007/SET007+455.ax').
%------------------------------------------------------------------------------
fof(fraenkel_a_4_0_closure3,axiom,
! [A,B,C,D,E] :
( ( m1_pboole(C,B)
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(B,C))) )
=> ( r2_hidden(A,a_4_0_closure3(B,C,D,E))
<=> ? [F] :
( m1_closure2(F,B,C,k6_closure2(B,C))
& A = k1_funct_1(F,E)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_4_4_closure3,axiom,
! [A,B,C,D,E] :
( ( m1_pboole(C,B)
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(B,C))) )
=> ( r2_hidden(A,a_4_4_closure3(B,C,D,E))
<=> ? [F] :
( m1_closure3(F,B,C,k6_closure2(B,C))
& A = k1_funct_1(F,E)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_5_1_closure3,axiom,
! [A,B,C,D,E,F] :
( ( m1_pboole(C,B)
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(B,C)))
& m1_subset_1(E,k1_zfmisc_1(k1_closure2(B,C))) )
=> ( r2_hidden(A,a_5_1_closure3(B,C,D,E,F))
<=> ? [G] :
( m1_closure3(G,B,C,k6_closure2(B,C))
& A = k1_funct_1(G,F)
& r2_hidden(G,k4_closure3(B,C,D,E)) ) ) ) ).
fof(dt_m1_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ! [D] :
( m1_closure3(D,A,B,C)
=> m1_pboole(D,A) ) ) ).
fof(existence_m1_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ? [D] : m1_closure3(D,A,B,C) ) ).
fof(redefinition_m1_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ! [D] :
( m1_closure3(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(reflexivity_r1_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> r1_closure3(A,B,D,D) ) ).
fof(reflexivity_r2_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> r2_closure3(A,B,C,C) ) ).
fof(dt_k1_closure3,axiom,
$true ).
fof(dt_k2_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> m4_pboole(k2_closure3(A,B,C),A,B) ) ).
fof(dt_k3_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> m1_subset_1(k3_closure3(A,B,C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ).
fof(commutativity_k3_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k3_closure3(A,B,C,D) = k3_closure3(A,B,D,C) ) ).
fof(idempotence_k3_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k3_closure3(A,B,C,C) = C ) ).
fof(redefinition_k3_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k3_closure3(A,B,C,D) = k2_xboole_0(C,D) ) ).
fof(dt_k4_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> m1_subset_1(k4_closure3(A,B,C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ).
fof(commutativity_k4_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k4_closure3(A,B,C,D) = k4_closure3(A,B,D,C) ) ).
fof(idempotence_k4_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k4_closure3(A,B,C,C) = C ) ).
fof(redefinition_k4_closure3,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
=> k4_closure3(A,B,C,D) = k3_xboole_0(C,D) ) ).
fof(dt_k5_closure3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( v11_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
& l1_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A))) ) ) ).
fof(fc1_closure3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ( v1_struct_0(g1_struct_0(u1_struct_0(A)))
& ~ v3_struct_0(g1_struct_0(u1_struct_0(A))) ) ) ).
fof(t1_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> k1_funct_4(B,C) = C ) ) ) ).
fof(t2_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r2_hidden(C,D)
=> r2_pboole(A,k6_mssubfam(A,B,k5_closure2(A,B,D)),C) ) ) ) ) ).
fof(t3_closure3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B))))
=> ( r1_tarski(C,k6_msualg_2(A,B))
=> ! [D] :
( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
=> ( r6_pboole(u1_struct_0(A),D,k6_mssubfam(u1_struct_0(A),u4_msualg_1(A,B),k5_closure2(u1_struct_0(A),u4_msualg_1(A,B),C)))
=> v3_msualg_2(D,A,B) ) ) ) ) ) ) ).
fof(d1_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r1_closure3(A,B,C,D)
<=> ! [E] :
~ ( r2_hidden(E,D)
& ! [F] :
~ ( r2_hidden(F,C)
& r1_tarski(E,F) ) ) ) ) ) ) ).
fof(d2_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r2_closure3(A,B,C,D)
<=> ! [E] :
~ ( r2_hidden(E,C)
& ! [F] :
~ ( r2_hidden(F,D)
& r1_tarski(F,E) ) ) ) ) ) ) ).
fof(t4_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
=> ( ( r1_closure3(A,B,D,C)
& r1_closure3(A,B,E,D) )
=> r1_closure3(A,B,E,C) ) ) ) ) ) ).
fof(t5_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
=> ( ( r2_closure3(A,B,C,D)
& r2_closure3(A,B,D,E) )
=> r2_closure3(A,B,C,E) ) ) ) ) ) ).
fof(d3_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( C = k1_closure3(A,B)
<=> C = a_2_0_closure3(A,B) ) ) ) ).
fof(t6_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> B = k1_funct_4(k1_pboole(A),k7_relat_1(B,k1_closure3(A,B))) ) ) ).
fof(t7_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ( ( k1_closure3(A,B) = k1_closure3(A,C)
& k7_relat_1(B,k1_closure3(A,B)) = k7_relat_1(C,k1_closure3(A,C)) )
=> r6_pboole(A,B,C) ) ) ) ) ).
fof(t8_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( ~ r2_hidden(C,k1_closure3(A,B))
=> k1_funct_1(B,C) = k1_xboole_0 ) ) ) ) ).
fof(t9_closure3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_closure2(C,A,B,k6_closure2(A,B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
~ ( r2_hidden(E,k1_funct_1(C,D))
& ! [F] :
( m1_closure2(F,A,B,k6_closure2(A,B))
=> ~ ( r2_hidden(E,k1_funct_1(F,D))
& v1_pre_circ(F,A)
& v1_finset_1(k1_closure3(A,F))
& r2_pboole(A,F,C) ) ) ) ) ) ) ) ).
fof(d4_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m4_pboole(D,A,B)
=> ( D = k2_closure3(A,B,C)
<=> ! [E] :
( r2_hidden(E,A)
=> k1_funct_1(D,E) = k3_tarski(a_4_0_closure3(A,B,C,E)) ) ) ) ) ) ).
fof(fc2_closure3,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ( v1_relat_1(k2_closure3(A,B,C))
& v3_relat_1(k2_closure3(A,B,C))
& v1_funct_1(k2_closure3(A,B,C))
& v1_pre_circ(k2_closure3(A,B,C),A) ) ) ).
fof(t10_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> r6_pboole(A,k2_closure3(A,B,C),k2_mboolean(A,k5_closure2(A,B,C))) ) ) ).
fof(t11_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> r6_pboole(A,k2_closure3(A,B,k3_closure3(A,B,C,D)),k2_pboole(A,k2_closure3(A,B,C),k2_closure3(A,B,D))) ) ) ) ).
fof(t12_closure3,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r1_tarski(C,D)
=> r2_pboole(A,k2_closure3(A,B,C),k2_closure3(A,B,D)) ) ) ) ) ).
fof(t13_closure3,conjecture,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> r2_pboole(A,k2_closure3(A,B,k4_closure3(A,B,C,D)),k3_pboole(A,k2_closure3(A,B,C),k2_closure3(A,B,D))) ) ) ) ).
%------------------------------------------------------------------------------